Q: A mechanic uses a wrench to loosen a bolt. If the wrench is 0.4 m long and the mechanic applies a force of 50 N perpendicular to the wrench at its end, what is the magnitude of the torque produced about the bolt?

$$\tau = 20\ \text{N}\cdot\text{m}$$

Q: A child pushes a merry-go-round with a tangential force $$F_{\text{tan}} at $$radius $$R. $$Which equation gives the work done by the child after the platform rotates through an angle $$\Delta\theta (in $$radians)?

$$W = F_{\text{tan}} R \Delta\theta$$

Q: A solid cylinder (mass $$50\ \text{kg}$$, radius $$0.06\ \text{m}) is $$pulled by a cable with force $$9.0\ \text{N}$$, as shown. What is the angular acceleration $$\alpha of $$the cylinder?

$$\alpha = \frac{F R}{I} = \frac{9.0 \times 0.06}{0.5 \times 50 \times 0.06^2$} = 6\ \text{rad/s}^2$$

Q: A bowling ball rolls without slipping down an incline at angle $$\beta. $$What is the acceleration of its center of mass?

$$a_{\text{cm}} = \frac{g \sin\beta}{1 + \frac{I}{MR^2$}}$$

Q: A grindstone (moment of inertia $$I = 2.0\ \text{kg}\cdot\text{m}^2) is $$spun by a motor exerting constant torque $$\tau = 10\ \text{N}\cdot\text{m}$$ for $$8.0\ \text{s}. $$What is the work done by the motor?

$$W = \tau \Delta\theta = \tau \frac{1}{2} \alpha t^2$

Question 1

A mechanic uses a wrench to loosen a bolt. If the wrench is 0.4 m long and the mechanic applies a force of 50 N perpendicular to the wrench at its end, what is the magnitude of the torque produced about the bolt?

Accepted Answer

$$	au = 20\ 	ext{N}\cdot	ext{m}$$

Question 2

A child pushes a merry-go-round with a tangential force $$F_{	ext{tan}} at $$radius $$R. $$Which equation gives the work done by the child after the platform rotates through an angle $$\Delta	heta (in $$radians)?

Accepted Answer

$$W = F_{	ext{tan}} R \Delta	heta$$

Question 3

A solid cylinder (mass $$50\ 	ext{kg}$$, radius $$0.06\ 	ext{m}) is $$pulled by a cable with force $$9.0\ 	ext{N}$$, as shown. What is the angular acceleration $$\alpha of $$the cylinder?

Accepted Answer

$$\alpha = \frac{F R}{I} = \frac{9.0 	imes 0.06}{0.5 	imes 50 	imes 0.06^2$} = 6\ 	ext{rad/s}^2$$

Question 4

Which scenario produces the greatest torque about point P for a rod, given the same force magnitude?

Accepted Answer

Force applied perpendicular at the farthest end from P

Question 5

A yo-yo consists of a solid disk of mass $$M$$ and radius $$R. If $$the tension in the string is $$T$$, what is the acceleration $$a of $$the yo-yo's center of mass?

Accepted Answer

$$a = \frac{g}{1 + \frac{I}{MR^2$}}$$

Question 6

A bowling ball rolls without slipping down an incline at angle $$\beta. $$What is the acceleration of its center of mass?

Accepted Answer

$$a_{	ext{cm}} = \frac{g \sin\beta}{1 + \frac{I}{MR^2$}}$$

Question 7

Which ranking correctly orders the angular accelerations $$\$$alpha_1$$, $$\$$alpha_2$$, $$\$$alpha_3$$ for the three cylinders shown, given the force application points?

Accepted Answer

$$\$$alpha_2$$ \u003e \$$alpha_1$$ \u003e \$$alpha_3$$

Question 8

A figure skater spins with arms extended, then pulls them in. What happens to her angular velocity and why?

Accepted Answer

Angular velocity increases because moment of inertia decreases, conserving angular momentum.

Question 9

A grindstone (moment of inertia $$I = 2.0\ 	ext{kg}\cdot	ext{m}^2) is $$spun by a motor exerting constant torque $$	au = 10\ 	ext{N}\cdot	ext{m}$$ for $$8.0\ 	ext{s}. $$What is the work done by the motor?

Accepted Answer

$$W = 	au \Delta	heta = 	au \frac{1}{2} \alpha t^2$

Question 10

Which equation correctly relates torque to the rate of change of angular momentum?

Accepted Answer

$$\sum 	au = \frac{dL}{dt}$$

Question 11

A mobile sculpture rotates in a horizontal plane. If the sum of external torques is zero, what happens to its angular momentum?

Accepted Answer

Angular momentum remains constant.

Question 12

A cyclist increases the speed of her bike wheels by pedaling harder. Which physical quantity is directly increased by her action?

Accepted Answer

Angular velocity $$\omega$$

Question 13

A jar lid is stuck. Which action maximizes the torque you can apply to open it?

Accepted Answer

Apply force perpendicular to the lid at the edge, maximizing the lever arm.

Question 14

A solid sphere rolls without slipping down a ramp. Which force is responsible for its rotation?

Accepted Answer

Static friction force $$f_s$$

Question 15

Which equation gives the angular momentum $$L of a $$rigid body rotating with angular velocity $$\omega$$ and moment of inertia $$I$$?

Accepted Answer

$$L = I \omega$$

Question 16

A spinning wheel slows down due to friction. Which statement best describes the change in angular momentum?

Accepted Answer

Angular momentum decreases because an external torque acts.

Question 17

A child applies a tangential force to a merry-go-round. Which equation gives the power delivered by the torque?

Accepted Answer

$$P = 	au \omega$$

Question 18

A figure skater spins faster by pulling her arms in. Which conservation law explains this phenomenon?

Accepted Answer

Conservation of angular momentum

Question 19

A rotating disk starts from rest and is acted upon by a constant torque. Which equation gives its angular velocity after time $$t$$?

Accepted Answer

$$\omega = \frac{	au}{I} t$$

Dynamics of Rotational Motion: Torque, Angular Acceleration, and Conservation Laws

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